Rms radius

Synthetic, nonionic polymers generally elute with little or no adsorption on TSK-PW columns. Characterization of these polymers has been demonstrated successfully using four types of on-line detectors. These include differential refractive index (DRI), differential viscometry (DV), FALLS, and MALLS detection (4-8). Absolute molecular weight, root mean square (RMS) radius of gyration, conformational coefficients, and intrinsic viscosity distributions have... 

The measurement of molecular weight and RMS radius provides the means to examine the conformational characteristics of a polymer using the relationship... 

The ionie strength of the mobile phase ean directly affect the elution volume and the RMS radius of a eationie polymer. Figure 20.14 is a plot of moleeular weight (caleulated with the Mini-Dawn detector) versus elution volume for a PVAm-HCl homopolymer (M = 95,000 Da) for three ionie strength levels... 

FIGURE 20.15 RMS radius versus ionic strength of cationic polymers and fully hydrolyzed PVA using CATSEC columns. 

The root mean square (rms) radius for the electron distribution is (3/5)mra and the corresponding rms effective diameter of the molecule is then given by ... 

Table IV. Adsorbed layer thickness 5 and the rms radius of gyration (S2) 5...

FIGURE 3.15 Standard plot of the log of the mean radium of gyration versus log molecular weight for differently shaped macromolecules. Essentially, for a sphere the radius is proportional to the root-mean-square (RMS) radius, and with a slope in the logrg versus log M of 1/3 for rod-shaped polymers, length is proportional to RMS radius and M with a slope of 1 and for random coils the end-to-end distance is proportional to the RMS radius and with a slope of about 0.5-0.6. 

Using the dipole form factor one can connect the third Zemach moment with the proton rms radius, and include the nuclear size correction of order Za) m in (7.62) on par with other contributions in (7.58) and (7.74) depending on the proton radius. Then the total dependence of the Lamb shift on Vp acquires the form [3, 25]... 

This expression also may be used for determination of the proton rms radius from the experimental data. Numerically it makes almost no difference because contributions in (7.63) and (7.64) with high accuracy coincide. However, the coefficient before r in (7.76) is modef dependent, so it is conceptuaffy advantageous to use the experimentaf vafue of the third Zemach moment obtained in [56] for cafculation of the nucfear size correction of order Za) m, and use the expression in (7.75) for determination of the proton radius from experimentaf data. 

Both the theoretical and experimental data for the classic 2S i/2 — 2Pi/2 Lamb shift are collected in Table 12.2. Theoretical results for the energy shifts in this Table contain errors in the parenthesis where the first error is determined by the yet uncalculated contributions to the Lamb shift, discussed above, and the second reflects the experimental uncertainty in the measurement of the proton rms charge radius. We see that the uncertainty of the proton rms radius is the largest source of error in the theoretical prediction of the classical Lamb shift. An immediate conclusion from the data in Table 12.2 is that the value of the proton radius [27] recently derived form the analysis of the world data on the electron-proton scattering seems compatible with the experimental data on the Lamb shift, while the values of the rms proton radius popular earlier [28, 29] are clearly too small to accommodate the experimental data on the Lamb shift. Unfortunately, these experimental results are rather widely scattered and have rather large experimental errors. Their internal consistency leaves much to be desired. 

Fig. 4.3-7. Comparison of different design strategies of high pressure cylindrical vessels A, Monobloc (eq. 4.3-9) B, Two piece shrink fit Rm = M-, R0 (eq. 4.3-12) C, Partially autof-rettaged cylinder (eq. 4.3-13) D, collapse pressure (pCOmPi. pi) of a Monobloc (eq. 4.3-10) a0,2 yield strength p imposed internal pressure, Rm radius of the shrink fit.

Another measure of the size of a polymer coil is the rms radius of gyration, Rq, defined by... 

Table I shows the MWs and the RMS radii of the PS standard determined with different dn/dc and A2 values. Equations 11 and 13 indicate that both the MW and the RMS radius increase as A2 increases. No significant changes were observed in MWs and sizes when A2 was varied at the constant dn/dc value of 0.2. It is noted that A2 could be assumed to be zero. As mentioned earlier, the product A cM is usually...

Figure 5 shows the plot of RMS radius versus MW on a log-log scale. The data for the elution volume lower than 3 mL were dropped... 

RMS radius versus elution volume for the PS standard shown... 

MWs and RMS radii of Perspex and UV 52E obtained from ThFFF-MALLS-RI are summarized in Table III. The values of dn/dc and A2 were taken as 0.083 and 2 X 10 , respectively, for both polymers. Figures 8 and 9 show the plots of MWD and the RMS radius versus... 

Figure 9. RMS radius versus MW for PMMA materials shown in Figure 6.

With its multiangle capability, MALLS can be used to measure the size as well as the absolute MW of polymers. The multiangle capability seems to be particularly important for the determination of MW and RMS radius of ultrahigh MW polymers as the Debye plot deviates from the linearity. It is also important for the analysis of ultrahigh MW polymers to use accurate A2 value as well as dn/dc as the resulting MW and RMS radius tend to vary with those values. 

Figure 3 shows the MAES-derived molar mass and rms radius as a function of elution volume for a broad polystyrene sample. Measurements were made in toluene at 690 nm. The value of dnidc chosen was 0.11. From Fig. 3, it should be noted that the radius data be-... 

Fig. 3 Molar mass and rms radius generated from data of Fig. 2 as a function of elution volume.

Figure 2.15. The three different monochromator/sample geometries used in powder diffraction a) flat diffracted beam monochromator, parallel arrangement b) curved diffracted beam monochromator, angular arrangement, and c) flat primary beam monochromator, parallel arrangement. F - focus of the x-ray source, S - sample, M - crystal monochromator, D - detector, Rm - radius of the monochromator focusing circle, Rq - radius of the goniometer focusing circle.

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